with \(x=(m_\mu /m_K)^2\) , \(y=(m_N/m_K)^2\)  and \(\lambda (1,x,y)=1+x^2+y^2-2(x+y+xy)\) .
The factor \(\rho _\mu (m_N)\)  increases from unity at \(m_N=0\)  to a maximum of 4.13 at \(m_N=263\)  MeV/\(c^2\) , and decreases to zero at the kinematic limit \(m_N=m_K-m_\mu \) . Assuming that the HNL decays exclusively to SM particles, its lifetime in the mass range \(m_N<m_K\)  exceeds \(10^{-4}/|U_4|^2~\mu \) s, where \(|U_4|^2\)  is the largest of the three coupling parameters \(|U_{\ell 4}|^2\)  (\(\ell =e,\mu ,\tau \) ) [1]}. Therefore under the above assumption, and additionally assuming conservatively that \(|U_{\ell 4}|^2<10^{-4}\) , the HNL can be considered stable in production-search experiments.
